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The integral points of an elliptic curve $E$ defined over $\mathbb Q$ are the points on a minimal model of $E$ that have integer coordinates.

To find them, pick a minimal Weierstrass equation for $E.$ The integral points are those solutions $(x,y)$ of the minimal Weierstrass equation that have integer coordinates. The number of integral points is finite (by a theorem of Siegel).

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